
Multivariate distributions deﬁned on the non-negative orthant are often not sufﬁciently ﬂexible: see, for example, the discussion on the multivariate Gamma in [1]}).
The only reasonable choice seems to be the multivariate log-Normal which, on the other hand, does not carry over to QML characteristics similarly to what was seen for the Gamma in the univariate case (as discussed in Section REF ).

Considering the one-to-one correspondence between the cumulative distribution function of an absolutely continuous \(K-\) dimensional random variable and its copula representation, resorting to copulas bypasses some issues: some proposals are in [2]}, where marginals for the components are linked together using Gaussian
or Student-T copulas.
However, there is no guidance for choices of a specific copula function except for convenience.
Moreover, using copulas in a vMEM context makes inference considerably complex.

It is not said that all components of the error term share the same marginal distribution.
Even though, in principle, one may be able to select the appropriate pdf one by one, this would require a lengthy model tuning (we will provide some evidence on our three volatility series below).

As a ﬁnal remark, the dynamics of the conditional mean is almost always the main focus of the analysis, so that a full ex ante speciﬁcation of the distribution of \(\mathbf {\varepsilon }_{t}\)  may be of secondary importance.

